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Complex Symmetry of Truncated Composition Operators

  • Project Type: Directed
  • Directed Project Contributors: Project Contributors: Ruth Jansen, Rebecca K. Rousseau

Purpose / Abstract

An N x N matrix is a representation of a linear function, called an operator, from RN to itself. For a given operator, each different basis representation of RN creates a different N x N matrix that represents that same operator. An operator is called complex symmetric if there is some basis for which the matrix representation is symmetric (possibly with both real- and complex-valued entries). This idea has also studied on infinite vector spaces such as the Hardy space and operators on those spaces, such as composition operators [2, 4]. Here, we define a truncated composition operator by considering only a finite-dimensional portion of a composition operator, so that we can represent it with a matrix and investigate any resulting complex symmetry.

Introduction / Background

Symmetric matrices, whose entries form a mirror image across the diagonal, are a familiar concept in the study of linear algebra and its applications. The concept of a complex symmetric operator is similar in nature and has been shown to have a wide variety of interesting applications within operator theory (Garcia, Putinar [2]). An operator is complex symmetric if it has a symmetric matrix representation with respect to some orthonormal basis. This is distinct from a self-adjoint matrix, in which entries across the diagonal are the complex conjugate of one another rather than identical.


Resources / Links

[1] Levon Balayan and Stephan Ramon Garcia. (2010). Unitary Equivalence to a Complex Symmetric Matrix: Geometric Criteria. Operators and Matrices, 4(1): 53-76.

[2] Stephen Ramon Garcia and Mihai Putinar. (2005). Complex Symmetric Operators and Applications. Transactions of the American Mathematical Society 358(3): 1285-1315.

[3] Carl Cowen and Barbara MacCluer. (1995). Composition Operators on Spaces of Analytic Functions. CRC Press, Inc.

[4] Sivaram K. Narayan, Daniel Sievewright, and Derek Thompson (2016). Complex Symmetric Composition Operators on H2. Journal of Mathematical Analysis and Applications, 443(1): 625-630.

[5] James E. Tener. (2010). Unitary Equivalence to a Complex Symmertric Matrix. Pomona College, Claremont, CA.